We extend the means of Discourse Representation Theory (DRT) and suggest representations of dialogues that keep track of the speech acts as events that are connected to each other via temporal relations and that can be qualified by predicates of a speech act typology. The contents of the speech acts are represented as flat, underspecified DRSs that are embedded in the global dialogue representation structure. Underspecified means that scope ambiguities are not resolved and flat means that the lexemes are assigned coarse-grained, ambiguity preserving representations. We develop a description language for the meaning of lexemes that provides guidelines for expanding the terms of the language to representations of different semantic granularity. We define coarse-grained representations of some typical expressions of the description language in terms of feature structures. We show how such representations can be incorporated in HPSG-analyses, how, using the Semantics Principle of HPSG, a compositional semantics for flat and underspecified representations can be defined, and how flat representations can be expanded to deeper representations. We sketch the model theory of flat, underspecified representations, point to some nice properties of the suggested HPSG-analyses for Machine Translation and outline the interplay of textual resolution procedures that seems to be needed for disambiguation in the dialogue context.